Convergence of functionals of sums of r.v.s to local times of fractional stable motions

Consider a sequence X_k=\sum_{j=0}^{\infty}c_j\xi_{k-j}, k\geq 1, where c_j, j\geq 0, is a sequence of constants and \xi_j, -\infty <j<\infty, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0<\alpha \leq 2. Let S_k=\sum_{j=1}^kX_j. Under suitable conditions on the constants c_j it is known that for a suitable normalizing constant \gamma_n, the partial sum process \gamma_n^{-1}S_{[nt]} converges in distribution to a linear fractional stable motion (indexed by \alpha and H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process X_k. In this paper it is established that the process n^{-1}\beta_n\sum_{k=1}^{[nt]}f(\beta_n(\gamma_n^{-1}S_k+x)) converges in distribution to (\int_{-\infty}^{\infty}f(y) dy)L(t,-x), where L(t,x) is the local time of the linear fractional stable motion, for a wide class of functions f(y) that includes the indicator functions of bounded intervals of the real line. Here \beta_n\to \infty such that n^{-1}\beta_n\to 0. The only further condition that is assumed on the distribution of \xi_1 is that either it satisfies the Cram\'er's condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000065

Limit Theorems for Functionals of Sums That Converge to Fractional Stable Motions

Too technical to post here. Please see paper.

Fixed points for nonexpansive mappings in Banach spaces

Bibliography: pages 140-142

Longevity of immediate rehabilitation with direct metal-wire reinforced composite fixed partial dentures.

OBJECTIVES This study aimed to analyze the longevity of direct metal-wire reinforced composite fixed partial dentures (MRC-FPD) and factors influencing their survival and success. METHODS Within one private practice 513 MRC-FPD were directly applied. The preparation of a proximal cavity in abutment teeth was not limited. MRC-FPD were reinforced by one to three metal-wires. At the last follow-up MRC-FPD were considered successful, if they were still in function without any need of therapy. MRC-FPD were considered as survived, if they were repaired or replaced. Multi-level Cox proportional hazard models were used to evaluate the association between clinical factors and time. RESULTS Mean follow-up period (range) was 59(2-249) months. Seventy-three bridges did not survive (cumulative survival rate(CSR):86%) and further 129 bridges had received a restorative follow-up treatment (CSR:61%). AFR was 2.2% for survival and 8.6% for success. In multivariate analysis MRC-FPD with> 1 wire showed a up to 2.3x higher failure rate than MRC-FPD with one wire(p ≤ 0.023). Dentist's experience in designing MRC-FDP (p ≤ 0.017), patient's caries risk (p ≤ 0.040) and bruxism (p = 0.033) significantly influenced the failure rate: the more experience, the lower caries risk and bruxism, the lower the failure rate. SIGNIFICANCE For directly prepared metal-wire reinforced composite bridges high survival and moderate success rates were observed. MRC-FPD might, thus, be an immediate, short- and medium-term solution for replacing missing teeth. However, several factors on the levels of practice (dentist's experience in designing MRC-FDP), patient (bruxism, caries risk) and restoration (number of wires) were identified as significant predictors for the failure rate. The study was registered in the German Clinical Trials Register (DRKS-ID: DRKS00021576)

Minimax Estimation of Nonregular Parameters and Discontinuity in Minimax Risk

When a parameter of interest is nondifferentiable in the probability, the existing theory of semiparametric efficient estimation is not applicable, as it does not have an influence function. Song (2014) recently developed a local asymptotic minimax estimation theory for a parameter that is a nondifferentiable transform of a regular parameter, where the nondifferentiable transform is a composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map. The contribution of this paper is two fold. First, this paper extends the local asymptotic minimax theory to nondifferentiable transforms that are a composite map of a Lipschitz continuous map having a finite set of nondifferentiability points and a translation-scale equivariant map. Second, this paper investigates the discontinuity of the local asymptotic minimax risk in the true probability and shows that the proposed estimator remains to be optimal even when the risk is locally robustified not only over the scores at the true probability, but also over the true probability itself. However, the local robustification does not resolve the issue of discontinuity in the local asymptotic minimax risk

Specific protease activity indicates the degree of Pseudomonas aeruginosa infection in chronic infected wounds

Chronic non-healing wounds are a major health problem with resident bacteria strongly implicated in their impaired healing. A rapid-screen to provide detailed knowledge of wound bacterial populations would therefore be of value and help prevent unnecessary and indiscriminate use of antibiotics—a process associated with promoting antibiotic resistance. We analysed chronic wound fluid samples, which had been assessed for microbial content, using 20 different fluorescent labelled peptide substrates to determine whether protease activity correlated with the bacterial load. Eight of the peptide substrates showed significant release of fluorescence after reaction with some of the wound samples. Comparison of wound fluid protease activities with the microbiological data indicated that there was no correlation between bacterial counts and enzyme activity for most of the substrates tested. However, two of the peptide substrates produced a signal corresponding with the microbial data revealing a strong positive correlation with Pseudomonas aeruginosa numbers. This demonstrated that short fluorescent labelled peptides can be used to detect protease activity in chronic wound fluid samples. The finding that two peptides were specific indicators for the presence of P. aeruginosa may be the basis for a diagnostic test to determine wound colonisation by this organism

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations